Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

203

borhood of a given point of the configuration, to what would
appear in Weierstrass's theory as x(t)dt, where t denotes the
parameter by means of which the neighborhood in question is
uniformized, and x(t) is analytic and does not vanish there.
Let P,(x) be an Abelian integral of the third kind with its
logarithmic discontinuities in the points x = =, x = 7, and let
Pen = P~(x) - P,(Y).
Moreover, Pj(x) shall be so chosen that
pxy_= p.
t~ xy
Klein defines his prime function Q(x1, x2; yl, Y2) as the following limit:
} _px+dz, y+dy
Q(x1, x2; Y1, y2) = lim \d dwyd e- xy
dx=O, dy=O
We can now state the definition of the prime function which
we propose to develop in detail in the following paragraphs. Let
the algebraic configuration be an arbitrary one of deficiency
p > 1, and let it be uniformized by automorphic functions with
limiting circle in the t-plane.* Let the integral P, transferred
to the t-plane, be written
tT *
Then
P t+At, T+AT
(1) 9(t, r) = lim /AtAre- t.
At=O, AT=0
In form, then, the definition is identical with Klein's.t But
whereas Klein's dwx is single-valued on a homogeneous configuration corresponding to the given algebraic configuration, our dt
is not invariant of the transformations of the automorphic group.
Transferred to the surface F it is infinitely multiple-valued.
On the other hand, Q(t, r) is a function of the two independent
variables t, r, each being chosen arbitrarily in the fundamental
* The details of this uniformization are set forth in the second edition of the
author's Funktionentheorie, vol. 1, 1912, ch. 14.
t Math. Ann., 36 (1889), p. 12. Cf. also Klein's account of the relation of his
prime function to Weierstrass's E(x, y) and Schottky's E(L, q); ibid., p. 13.
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